# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Corollary 7.4.5.5. Let $\operatorname{\mathcal{C}}$ be a small simplicial set. Then any diagram $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ admits a colimit in the $\infty$-category $\operatorname{\mathcal{S}}$. Moreover, a Kan complex $X$ is a colimit of the diagram $\mathscr {F}$ if and only if there exists a weak homotopy equivalence $\int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow X$.

Proof. Apply Corollary 7.4.5.4 to the left fibration $U: \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ of Example 5.7.2.9. $\square$